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A090319 Fifth column (k=4) of triangle A084938. 2
1, 4, 14, 52, 217, 1040, 5768, 36992, 272584, 2285184, 21550656, 226071744, 2611146384, 32911082496, 449243785728, 6598780563456, 103734755882496, 1737181702840320, 30866291090657280, 579859321408266240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..150

FORMULA

a(n) = Sum_{k=0..n} A090595(k)*(n-k)!.

a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!.

a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k).

G.f.: (Sum_{k>=0} k!*x^k)^4.

From G. C. Greubel, Dec 29 2019: (Start)

a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function.

a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End)

MAPLE

seq( (n+3)!*add(add(add( Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019

MATHEMATICA

Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k, 0, n}, {m, 0, k}, {j, 0, m}], {n, 0, 20}] (* G. C. Greubel, Dec 29 2019 *)

PROG

(PARI) vector(21, n, my(b=binomial); sum(k=0, n-1, sum(m=0, k, sum(j=0, m, (n-1)!/(b(n-1, k)*b(k, m)*b(m, j)) )))) \\ G. C. Greubel, Dec 29 2019

(MAGMA) F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n, k)*B(k, m)*B(m, j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

(Sage) b=binomial; [sum(sum(sum(factorial(n)/(b(n, k)*b(k, m)*b(m, j)) for j in (0..m)) for m in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

(GAP) B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], m-> Sum([0..m], j-> Factorial(n)/(B(n, k)*B(k, m)*B(m, j)) )))); # G. C. Greubel, Dec 29 2019

CROSSREFS

Cf. A084938.

Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595.

Sequence in context: A308023 A149489 A125783 * A295518 A047118 A017948

Adjacent sequences:  A090316 A090317 A090318 * A090320 A090321 A090322

KEYWORD

easy,nonn

AUTHOR

Philippe Deléham, Feb 05 2004

STATUS

approved

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Last modified September 22 17:01 EDT 2021. Contains 347607 sequences. (Running on oeis4.)